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Theorem bnj1452 30820
 Description: Technical lemma for bnj60 30830. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1452.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1452.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1452.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1452.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1452.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1452.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1452.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1452.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1452.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1452.10 𝑃 = 𝐻
bnj1452.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1452.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1452.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1452.14 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
Assertion
Ref Expression
bnj1452 (𝜒𝐸𝐵)
Distinct variable groups:   𝐴,𝑑,𝑥,𝑧   𝐸,𝑑,𝑧   𝑅,𝑑,𝑥,𝑧   𝜒,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑦,𝑓)   𝐵(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑅(𝑦,𝑓)   𝐸(𝑥,𝑦,𝑓)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1452
StepHypRef Expression
1 bnj1452.14 . . 3 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
2 bnj1452.5 . . . . . 6 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
3 bnj1452.7 . . . . . 6 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
42, 3bnj1212 30570 . . . . 5 (𝜒𝑥𝐴)
54snssd 4314 . . . 4 (𝜒 → {𝑥} ⊆ 𝐴)
6 bnj1147 30762 . . . . 5 trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴
76a1i 11 . . . 4 (𝜒 → trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴)
85, 7unssd 3772 . . 3 (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ⊆ 𝐴)
91, 8syl5eqss 3633 . 2 (𝜒𝐸𝐴)
10 elsni 4170 . . . . . . . 8 (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
1110adantl 482 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → 𝑧 = 𝑥)
12 bnj602 30685 . . . . . . 7 (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
1311, 12syl 17 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
14 bnj1452.6 . . . . . . . . . 10 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
1514simplbi 476 . . . . . . . . 9 (𝜓𝑅 FrSe 𝐴)
163, 15bnj835 30529 . . . . . . . 8 (𝜒𝑅 FrSe 𝐴)
17 bnj906 30700 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
1816, 4, 17syl2anc 692 . . . . . . 7 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
1918ad2antrr 761 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
2013, 19eqsstrd 3623 . . . . 5 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
21 ssun4 3762 . . . . . 6 ( pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅) → pred(𝑧, 𝐴, 𝑅) ⊆ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2221, 1syl6sseqr 3636 . . . . 5 ( pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
2320, 22syl 17 . . . 4 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
2416ad2antrr 761 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
25 simpr 477 . . . . . . . 8 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
266, 25bnj1213 30569 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧𝐴)
27 bnj906 30700 . . . . . . 7 ((𝑅 FrSe 𝐴𝑧𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
2824, 26, 27syl2anc 692 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
294ad2antrr 761 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑥𝐴)
30 bnj1125 30760 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3124, 29, 25, 30syl3anc 1323 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3228, 31sstrd 3598 . . . . 5 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3332, 22syl 17 . . . 4 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
341bnj1424 30609 . . . . 5 (𝑧𝐸 → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3534adantl 482 . . . 4 ((𝜒𝑧𝐸) → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3623, 33, 35mpjaodan 826 . . 3 ((𝜒𝑧𝐸) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
3736ralrimiva 2965 . 2 (𝜒 → ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
38 snex 4874 . . . . . . . 8 {𝑥} ∈ V
3938a1i 11 . . . . . . 7 (𝜒 → {𝑥} ∈ V)
40 bnj893 30698 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → trCl(𝑥, 𝐴, 𝑅) ∈ V)
4116, 4, 40syl2anc 692 . . . . . . 7 (𝜒 → trCl(𝑥, 𝐴, 𝑅) ∈ V)
4239, 41bnj1149 30563 . . . . . 6 (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ∈ V)
431, 42syl5eqel 2708 . . . . 5 (𝜒𝐸 ∈ V)
44 bnj1452.1 . . . . . 6 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
4544bnj1454 30612 . . . . 5 (𝐸 ∈ V → (𝐸𝐵[𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)))
4643, 45syl 17 . . . 4 (𝜒 → (𝐸𝐵[𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)))
47 bnj602 30685 . . . . . . . 8 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
4847sseq1d 3616 . . . . . . 7 (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
4948cbvralv 3164 . . . . . 6 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
5049anbi2i 729 . . . . 5 ((𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
5150sbcbii 3478 . . . 4 ([𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ [𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
5246, 51syl6bb 276 . . 3 (𝜒 → (𝐸𝐵[𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)))
53 sseq1 3610 . . . . . 6 (𝑑 = 𝐸 → (𝑑𝐴𝐸𝐴))
54 sseq2 3611 . . . . . . 7 (𝑑 = 𝐸 → ( pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
5554raleqbi1dv 3140 . . . . . 6 (𝑑 = 𝐸 → (∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
5653, 55anbi12d 746 . . . . 5 (𝑑 = 𝐸 → ((𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
5756sbcieg 3455 . . . 4 (𝐸 ∈ V → ([𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
5843, 57syl 17 . . 3 (𝜒 → ([𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
5952, 58bitrd 268 . 2 (𝜒 → (𝐸𝐵 ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
609, 37, 59mpbir2and 956 1 (𝜒𝐸𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1992  {cab 2612   ≠ wne 2796  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3191  [wsbc 3422   ∪ cun 3558   ⊆ wss 3560  ∅c0 3896  {csn 4153  ⟨cop 4159  ∪ cuni 4407   class class class wbr 4618  dom cdm 5079   ↾ cres 5081   Fn wfn 5845  ‘cfv 5850   predc-bnj14 30453   FrSe w-bnj15 30457   trClc-bnj18 30459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-reg 8442  ax-inf2 8483 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-1o 7506  df-bnj17 30452  df-bnj14 30454  df-bnj13 30456  df-bnj15 30458  df-bnj18 30460  df-bnj19 30462 This theorem is referenced by:  bnj1312  30826
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